Residues and Duality for Projective Algebraic Varieties

The main objective of these lecture notes is to describe local and global duality, primarily focusing on irreducible algebraic varieties over an algebraically closed field. Both local and global duality theorems are based on two linear operators - we have the residue map defined on the top local cohomology of the canonical sheaf, and the integral as a linear form on the top global sheaf cohomology of algebraic variety. There is then a comparison given by the residue theorem relating the integral operator to a sum of residues or, when specializing to projective algebraic curves, yielding the Serre duality theorem expressed in terms of differentials and their residues. Possible applications include the generalization of residue calculus to toric residues.